A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

8, 22, 27, 32, 77, 125, 128, 243, 343, 494, 512, 611, 660, 1073, 1281, 1331, 1425, 2033, 2048, 2187, 2197, 2332, 3125, 4172, 4565, 4913, 5293, 6031, 6859, 8192, 9983, 12167, 13969, 15818, 15947, 16807, 17485, 19683, 23489, 23840, 24389, 25241, 25389, 29791, 32768

Offset

1,1

Comments

Semiprimes in the sequence: 22, 77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 98099, 99101, .... - Robert Israel, Mar 29 2018

2^k is a term for all odd k > 1. - Michael S. Branicky, Aug 22 2021

Links

Amiram Eldar, Table of n, a(n) for n = 1..1009 (terms below 10^9, terms 1..100 from Paolo P. Lava)

Example

Aliquot parts of 77 are 1, 7, 11. Then (1^2 + 7^2 + 11^2)/(1 + 7 + 11) = 171/19 = 9.

Maple

with(numtheory): P:=proc(n)
if not isprime(n) and frac((add(p^2, p=divisors(n))-n^2)/(sigma(n)-n))=0
then n; fi; end: seq(P(i), i=2..35*10^3);

Mathematica

aQ[n_] := CompositeQ[n] && Divisible[DivisorSigma[2, n] - n^2, DivisorSigma[1, n] - n]; Select[Range[33000], aQ] (* Amiram Eldar, Aug 17 2019 *)

Prog

(PARI) isok(n) = (n!=1) && !isprime(n) && (((sigma(n, 2) - n^2) % (sigma(n) - n)) == 0); \\ Michel Marcus, Mar 23 2018
(Python)
from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return len(divs) > 1 and sum(d**2 for d in divs)%sum(divs) == 0
print(list(filter(ok, range(4, 32769)))) # Michael S. Branicky, Aug 22 2021

Crossrefs

Cf. A001065, A001157, A020487, A067558.

Contains A056824.

Sequence in context: A304301 A236917 A064193 * A213017 A305515 A360510

Adjacent sequences: A301479 A301480 A301481 * A301483 A301484 A301485

Keyword

nonn,easy

Author

Paolo P. Lava, Mar 22 2018

Status

approved